"Arab" or "Arabic" or "Arabian"?
As adjectives, the New Oxford American Dictionary defines Arabic a related to the language or the literature, while Arab is the more generic term (“of or relating to Arabia and the people of Arabia”) and Arabian is a historical variant of Arab.
Some dictionaries (including WordNet) report a wider meaning of Arabic as being roughly synonymous with Arab and Arabian. In some established expressions, like “arabic coffee”, arabic is favoured. Note, however, that usage favours “arab cuisine”, for example!
I would use "Arabic" of the language and literature, and "Arab" otherwise. I wouldn't use "Arabian" except perhaps referring to the the country "Saudi Arabia" - but I would more likely say "Saudi".
I would however use "Arabian" in set historical phrases like The Arabian Nights.
These words are not interchangeable.
Arab: relates to the people, places and culture of the Arab world. Examples include: Arab cuisine, the Arab League, Arabs in the Diaspora, Arab influence in Western Music, etc.
Arabic: specifically relates to the written and spoken language of the Arab world. Examples: Arabic literature, Arabic influences in English, dialects of Arabic, etc.
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Arabian: historic or literary in its relation to people or things of the Arab world. Examples: the Arabian nights, under the Arabian moonlight. A note on usage from NOAD:
Arab is now generally used in reference to people; the use of Arabian in this sense is historical
Current usage of the word, however, is found in these examples: Arabian Peninsula, Arabian Sea and Arabian [horse]. Also, the demonym for Saudi Arabia is either Saudi or Saudi Arabian, not Arabian.
I initially thought Arabic coffee did not exist. Coffee arabica or arabica coffee is a major commercial species grown in East Africa and Latin America (originated in Ethiopia and Yemen). Ironically, the Arab world is not a major producer of the commodity. Yemen is the only notable Arab nation that grows coffee (the arabica variety), but it is not a major producer. However, it seems Arabic coffee is a special exception to the rule. Credit goes to fellow user, Amanda, for enlightening me.